Curvature of Time?

We can all see what curvature of space looks like, just by throwing a ball and watching it follow the natural geodesic.

But what does curvature of time look like?

How do we experience it?

We typically experience the passage of time in what seems to be a forward linear manner. The forward part seems to be due to how our nervous system works, thus giving a chronological bias towards causality in our perception.

But if we can see how gravity curves space, then how do we percieve how it affects time?

We can all see what curvature of space looks like, just by throwing a ball and watching it follow the natural geodesic.

Objects in GR don't generally follow geodesics in space, they follow geodesics in spacetime, usually the paths that minimize the proper time (although I gather it can maximize it in certain cases). I'm pretty sure a ball isn't following a geodesic in space when you throw it (unless you're in flat spacetime and the ball goes in a straight line).

We can all see what curvature of space looks like, just by throwing a ball and watching it follow the natural geodesic.

Sorry, that is not the curvature of space.

If you throw a ball into the air and then someone throws you into the air as well, you'll see the ball moving in a straight line relative to yourself (in the absence of air resistance, of course).

At least, that's what happens at first. If both objects remain in free-fall long enough, eventually the ball will start to change course slightly (or speed up or slow down), due to the fact that the acceleration due to gravity is not constant everywhere. Now that's the curvature of space-time.

We can all see what curvature of space looks like, just by throwing a ball and watching it follow the natural geodesic.

No. The trajectory of the ball is mainly an effect of "curved time". Curved space produces only minor effects like orbit precession and additional light bending (doubling the amount caused by "curved time" alone). But note that "curved time" is not possible, without curved space, because you cannot have only one dimension of a manifold curved. So it's best to talk about curved spacetime.

To say it another way, if the curvature of the ball's path were representative of the curvature of space, then space would be mighty curved right there, and we'd have thrown out Euclidean Geometry centuries ago. In other words, if that parabolic path represented a straight line in that region of space, then there's no way the Pythagorean Theorem would hold in that region.

As others have said, the geodesic lies in the 4-d space-time surface (manifold).